Question

A derivative formula a. Use the definition of the derivative to determine \(\frac{d}{d x}\left(a x^{2}+b x+c\right),\) where \(a, b,\) and \(c\) are constants. b. Let \(f(x)=4 x^{2}-3 x+10\) and use part (a) to find \(f^{\prime}(x)\) c. Use part (b) to find \(f^{\prime}(1)\)

Short answer

The derivative of the quadratic function \(f(x) = 4x^2 - 3x + 10\) is \(f'(x) = 8x - 3\), and its value at \(x = 1\) is \(f'(1) = 5\).

Step-by-step solution

Definition of the derivative

Recall the definition of derivative: $$\frac{d}{d x} f(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$ For the given quadratic function \(f(x) = ax^2 + bx + c\), we need to find the derivative using this definition.

Evaluate the function at \((x+h)\)

Determine \(f(x + h)\). Replace \((x)\) with \((x + h)\) in the equation: \(f(x + h) = a(x + h)^2 + b(x + h) + c\) Expand the expressions in the equation: \(f(x + h) = a(x^2 + 2hx + h^2) + b(x + h) + c\) Simplify the equation: \(f(x + h) = ax^2 + 2ahx + ah^2 + bx + bh + c\)

Apply the derivative definition

Now, we apply the definition of derivative: \(\frac{d}{d x} (ax^2 + bx + c) = \lim_{h \to 0} \frac{[ax^2 + 2ahx + ah^2 + bx + bh + c] - [ax^2 + bx + c]}{h}\) Simplify the expression: \(\frac{d}{d x} f(x) = \lim_{h \to 0} \frac{2ahx + ah^2 + bh}{h}\)

Finalize the derivative

Cancel out \(h\) in the expression above: \(\frac{d}{d x} f(x) = \lim_{h \to 0} [2ax + ah + b]\) As \(h \to 0\), the expression becomes: \(\frac{d}{d x} f(x) = 2ax + b\) Now, we have the derivative of the quadratic function.

Apply the formula to the given function

We need to find the derivative of \(f(x) = 4x^2 - 3x + 10\). Using our derived formula \(\frac{d}{d x} f(x) = 2ax + b\), plug in \(a = 4\) and \(b = -3\): \(f'(x) = 2(4)x - 3\) \(f'(x) = 8x - 3\)

Evaluate the derivative at \(x = 1\)

We are asked to find \(f'(1)\). Plug in \(x = 1\) in the equation \(f'(x) = 8x - 3\): \(f'(1) = 8(1) - 3\) \(f'(1) = 5\) Therefore, \(f'(1) = 5\).

Key concepts

Definition of the Derivative

Understanding the derivative of a function is crucial in calculus. A derivative represents how a function changes as its input changes. In a graphical sense, it gives the slope of the function at any given point.

Mathematically, the derivative of a function at a point is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. The formal definition uses the limit process, expressed as: $$\frac{d}{dx} f(x) = \lim_{{h \to 0}} \frac{f(x + h) - f(x)}{h}$$This formula captures the essence of the derivative as the instantaneous rate of change, mirroring the way speed at any moment is the instantaneous rate of change of distance with respect to time.

Limit Process in Calculus

The limit process is a foundational concept in calculus that deals with finding values that functions approach as the variables approach a specific point. Limits help us understand the behavior of functions near a point, even if the function is not defined at that exact point.

The notation \( \lim_{{h \to 0}} \) signifies that we are looking at what happens to our expression as \( h \) gets infinitesimally small, essentially zero but not quite there. In the context of the derivative, the limit process allows us to evaluate the rate of change at an exact point, rather than over an interval, by considering what the average rate of change approaches as the interval becomes very small.

Simplifying Algebraic Expressions

The simplification of algebraic expressions is an important skill when working with derivatives. Breaking down and combining like terms makes it possible to identify patterns and apply the rules of differentiation effectively.

In the exercise, after replacing \( x \) with \( x + h \), we've expanded and simplified the expression before taking the limit. Simplifying involves expanding polynomials, collecting like terms, and canceling where applicable—this is critical to ensure that the final derivative can be easily computed, especially when the h terms cancel out after applying the limit as \( h \) approaches zero.

Applying Derivatives

Applying derivatives is the practical side of differentiation, involving the use of derivative formulas to solve real problems. It's where theoretical knowledge meets real-world application, often in physics, engineering, economics, and other sciences.

In the given exercise, after finding the general formula for the derivative of a quadratic function, we applied it to a specific function \( f(x) = 4x^2 - 3x + 10 \). By substituting the specific coefficients into the general derivative formula, we quickly found the slope of the function at any point \( x \), denoted as \( f'(x) \). This allows us to evaluate the behavior and rate of change of the function at any given point, for example finding \( f'(1) \) which is the slope of the function when \( x = 1 \). Derivatives also provide critical information for optimizing functions, finding maximum and minimum points, and solving problems related to rates of change.